Finite Group Databasebeta

← #13150314

Subgroups — #13150314

Order 1728.

742 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721819C211yesmaximal, characteristic
2C9×#926C211yesmaximal, characteristic, complemented
3C3×#63486C211yesmaximal, characteristic, complemented
4#11721275C212yesmaximal, complemented
5#11721242C212yesmaximal, complemented
6#11720994C212yesmaximal, complemented
7#11720993C211yesmaximal, characteristic, complemented
8#11720970C211yesmaximal, characteristic, complemented
9#11720952C211yesmaximal, characteristic, complemented
10#11720920C212yesmaximal, complemented
11Dic3×Dic18C211yesmaximal, characteristic, complemented
12#10594814—31nomaximal
13#10589636—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16C2×C4⋊(C3⋊D9)C2211yescharacteristic
17C2×C3×Dic18C2211yescharacteristic
18C2×(C3⋊D9)⋊C4C2212yes—
19C2×#6189C2212yes—
20#88536C2212yes—
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22C9×#217C2211yescharacteristic
23C9×#217C2211yescharacteristic
24C3×#3676C2212yes—
25C9×C4⋊Dic3C2211yescharacteristic
26C9×C4⋊Dic3C2211yescharacteristic
27C4×C9×Dic3C2211yescharacteristic
28C4×C9×Dic3C2211yescharacteristic
29C3×C4⋊Dic9C2211yescharacteristic
30C3×Dic9⋊C4C2212yes—
31#88445C2212yes—
32#88445C2212yes—
33#88445C2212yes—
34Dic3⋊Dic9C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37#64139—31no—
38C3×#926D311yescharacteristic
39C3×#924—31no—
40#63975—32no—
41#63953—32no—
42#63918—31no—
43#63895—32no—
44#63894—31no—
45#63864—31no—
46#63848—32no—
47Dic3×Dic6—31no—
48C9×#117—31no—
49#63492—31no—
50#63492—61no—
51#63486D311yescharacteristic, complemented
52#63460—32no—
53#63458—32no—
54#63447—32no—
55#63445—32no—
56#63437—31no—
57C4×D36—31no—
58#63431—31no—
59C4×Dic18—31no—
60C23×C3×C9C2311yescharacteristic
61C22×C3⋊D9C2312yes—
62C2×C3×C4×C9—22no—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65#6233—42no—
66C2×C3⋊Dic9C2312yes—
67C4⋊(C3⋊D9)—22no—
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C9×Dic3C2311yescharacteristic
72C2×C3×Dic9C2312yes—
73C2×C3×Dic9C2312yes—
74C3×Dic18—22no—
75(C3⋊D9)⋊C4—42no—
76#6189—42no—
77#5643—91no—
78#5534—91no—
79C22×C32×C4C2×D311yescharacteristic
80C2×#638—32no—
81C2×C4⋊(C3⋊D3)—31no—
82C2×C3×Dic6—31no—
83C2×#627—32no—
84C2×#624—32no—
85#3752—32no—
86C4⋊(C3⋊Dic3)—31no—
87C3×#217—32no—
88C3×#217C2×D311yescharacteristic
89C3×#217C2×D311yescharacteristic
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×Dic3⋊C4—32no—
94C3×C4×Dic3C2×D311yescharacteristic
95C3×C4×Dic3C2×D311yescharacteristic
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—32no—
98#3722—32no—
99#3722—32no—
100#3722—32no—
101Dic32—32no—
102C22×C4×C9—21no—
103C22×C4×C9C2×D311yescharacteristic
104C2×#611—32no—
105C2×#611—62no—
106C2×D36—31no—
107C2×D36—61no—
108C2×C4×D9—32no—
109C2×Dic18—32no—
110C2×Dic18C2×D311yescharacteristic
111C9×C4⋊C4—31no—
112C9×C4⋊C4—31no—
113C9×C22⋊C4—31no—
114C9×C22⋊C4—31no—
115C42×C9—31no—
116C42×C9—31no—
117#3676C2×D312yes—
118(C2×D9)⋊C4—32no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4—32no—
121(C2×D9)⋊C4—32no—
122(C2×D9)⋊C4—62no—
123C4⋊Dic9—31no—
124C4⋊Dic9—32no—
125C4⋊Dic9—61no—
126C4⋊Dic9C2×D311yescharacteristic
127Dic9⋊C4—32no—
128Dic9⋊C4C2×D312yes—
129C4×Dic9—32no—
130C22×C3×C9—21no—
131C22×C3×C9—21no—
132C22×C3×C9—21no—
133C22×C3×C9C2411yescharacteristic
134C2×C3⋊D9—22no—
135C2×C3⋊D9—22no—
136C2×C3⋊D9—22no—
137C3×C4×C9—21no—
138C3×C4×C9#4112yes—
139C3⋊Dic9—22no—
140C9×Dic3—21no—
141C9×Dic3—21no—
142C9×Dic3—21no—
143C9×Dic3—21no—
144C3×Dic9—22no—
145C3×Dic9—22no—
146C3×C22⋊Q8—91no—
147C3×#117—31no—
148#947—91no—
149Q8×Dic3—91no—
150#930—91no—
151#930—91no—
152#930—181no—
153#926D911yescharacteristic, complemented
154#924—31no—
155#898—92no—
156#898—92no—
157#897—92no—
158#896—92no—
159#893—91no—
160#888—91no—
161#885—92no—
162#885—92no—
163#883—92no—
164#883—92no—
165#875—91no—
166C4×D12—91no—
167#869—91no—
168C4×Dic6—91no—
169C23×C32C22×D311yescharacteristic
170C22×C3⋊D3—32no—
171C2×C32×C4—22no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174#638—122no—
175C2×C3⋊Dic3—32no—
176C4⋊(C3⋊D3)—62no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3C22×D311yescharacteristic
180C2×C3×Dic3C22×D311yescharacteristic
181C2×C3×Dic3C22×D311yescharacteristic
182C2×C3×Dic3C22×D311yescharacteristic
183C3×Dic6—62no—
184#627—122no—
185#624—122no—
186C23×C9—21no—
187C23×C9C22×D311yescharacteristic
188C22×D9—32no—
189C22×D9—62no—
190C2×C4×C9—22no—
191C2×C4×C9—21no—
192C2×C4×C9—21no—
193C2×C4×C9—31no—
194C2×C4×C9—31no—
195C2×C4×C9—31no—
196C2×C4×C9—31no—
197C2×C4×C9—42no—
198C2×C4×C9C22×D311yescharacteristic
199C2×C4×C9C22×D311yescharacteristic
200#611—122no—
201#611—124no—
202C2×Dic9—32no—
203C2×Dic9—62no—
204C2×Dic9C22×D312yes—
205C2×Dic9C22×D312yes—
206D36—62no—
207D36—122no—
208C4×D9—122no—
209Dic18—22no—
210Dic18—122no—
211#542—271noSylow
212C2×C3×C9—41no—
213C2×C3×C9C2×#4111yescharacteristic
214C2×C3×C9H(5,𝔽2)11yescharacteristic
215C2×C3×C92−1+411yescharacteristic
216C3⋊D9—42no—
217C2×C3×Q8—91no—
218C22×C3×C4—21no—
219C22×C3×C4C2×D911yescharacteristic
220C22×C3×C4D3211yescharacteristic
221C2×#67—92no—
222C2×#67—92no—
223C2×#67—182no—
224C2×D12—91no—
225C2×D12—91no—
226C2×D12—181no—
227C2×C4×D3—92no—
228C2×C4×D3—92no—
229C2×Dic6—31no—
230C2×Dic6—92no—
231C2×Dic6—92no—
232C3×C4⋊C4—31no—
233C3×C4⋊C4—31no—
234C3×C4⋊C4—91no—
235C3×C4⋊C4—92no—
236C3×C22⋊C4—31no—
237C3×C22⋊C4—31no—
238C3×C22⋊C4—92no—
239C3×C42—31no—
240C3×C42—31no—
241#217—32no—
242#217C2×D911yescharacteristic
243#217C2×D911yescharacteristic
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—92no—
250(C2×D3)⋊C4—92no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—182no—
253C4⋊Dic3—31no—
254C4⋊Dic3—92no—
255C4⋊Dic3—91no—
256C4⋊Dic3—91no—
257C4⋊Dic3—92no—
258C4⋊Dic3—181no—
259C4⋊Dic3C2×D911yescharacteristic
260C4⋊Dic3C2×D911yescharacteristic
261Dic3⋊C4—32no—
262Dic3⋊C4—92no—
263Dic3⋊C4—92no—
264C4×Dic3—92no—
265C4×Dic3—92no—
266C4×Dic3C2×D911yescharacteristic
267C4×Dic3C2×D911yescharacteristic
268C22×C32—21no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32C23×D311yescharacteristic
272C2×C3⋊D3—62no—
273C2×C3⋊D3—62no—
274C2×C3⋊D3—62no—
275C32×C4—21no—
276C32×C4#23512yes—
277C3⋊Dic3—62no—
278C3×Dic3—21no—
279C3×Dic3—21no—
280C3×Dic3—21no—
281C3×Dic3—21no—
282C3×Dic3—62no—
283C3×Dic3—62no—
284C22×C9—22no—
285C22×C9—21no—
286C22×C9—21no—
287C22×C9—21no—
288C22×C9—21no—
289C22×C9—41no—
290C22×C9—41no—
291C22×C9C23×D311yescharacteristic
292C2×D9—62no—
293C2×D9—62no—
294C2×D9—62no—
295C2×D9—122no—
296C2×D9—122no—
297C2×D9—122no—
298C4×C9—21no—
299C4×C9—22no—
300C4×C9—41no—
301C4×C9—61no—
302C4×C9—61no—
303C4×C9—61no—
304C4×C9—61no—
305C4×C9#23712yes—
306Dic9—22no—
307Dic9—22no—
308Dic9—62no—
309Dic9—122no—
310C4⋊Q8—271no—
311#126—272no—
312#124—271no—
313#124—272no—
314#123—272no—
315C22⋊Q8—91no—
316C22⋊Q8—272no—
317#121—271no—
318C4×Q8—271no—
319C4×D4—271no—
320#117—31no—
321C3×C9#54211yescharacteristic, complemented, Sylow
322C23×C3—21no—
323C23×C3C22×D911yescharacteristic
324C23×C3C2×D3211yescharacteristic
325C22×D3—92no—
326C22×D3—92no—
327C22×D3—182no—
328C3×Q8—182no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—31no—
334C2×C3×C4—31no—
335C2×C3×C4—31no—
336C2×C3×C4—31no—
337C2×C3×C4—42no—
338C2×C3×C4—92no—
339C2×C3×C4—92no—
340C2×C3×C4C22×D911yescharacteristic
341C2×C3×C4C22×D911yescharacteristic
342C2×C3×C4C2×D3211yescharacteristic
343C2×C3×C4C2×D3211yescharacteristic
344#67—362no—
345#67—362no—
346#67—364no—
347C2×Dic3—32no—
348C2×Dic3—32no—
349C2×Dic3—92no—
350C2×Dic3—92no—
351C2×Dic3—182no—
352C2×Dic3C22×D911yescharacteristic
353C2×Dic3C22×D911yescharacteristic
354C2×Dic3C22×D911yescharacteristic
355C2×Dic3C22×D911yescharacteristic
356D12—182no—
357D12—182no—
358D12—362no—
359C4×D3—362no—
360C4×D3—362no—
361Dic6—62no—
362Dic6—362no—
363Dic6—362no—
364C2×C32—41no—
365C2×C32C2×#23511yescharacteristic
366C2×C32#100911yescharacteristic
367C2×C32#101011yescharacteristic
368C3⋊D3—122no—
369C2×C9—21no—
370C2×C9—21no—
371C2×C9—21no—
372C2×C9—42no—
373C2×C9—41no—
374C2×C9C2×#23711yescharacteristic
375C2×C9#100711yescharacteristic
376C2×C9#100911yescharacteristic
377D9—122no—
378D9—242no—
379C2×Q8—91no—
380C2×Q8—272no—
381C2×D4—272no—
382C2×D4—271no—
383C22×C4—272no—
384C22×C4D3×D911yescharacteristic
385C4⋊C4—31no—
386C4⋊C4—31no—
387C4⋊C4—91no—
388C4⋊C4—92no—
389C4⋊C4—272no—
390C4⋊C4—272no—
391C4⋊C4—271no—
392C22⋊C4—31no—
393C22⋊C4—31no—
394C22⋊C4—92no—
395C22⋊C4—272no—
396C22⋊C4—272no—
397C22⋊C4—272no—
398C22⋊C4—272no—
399C42—31no—
400C42—31no—
401C42—272no—
402C22×C3—22no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—41no—
411C22×C3—41no—
412C22×C3C23×D911yescharacteristic
413C22×C3C22×D3211yescharacteristic
414C2×D3—182no—
415C2×D3—182no—
416C2×D3—182no—
417C2×D3—182no—
418C2×D3—182no—
419C2×D3—182no—
420C2×D3—362no—
421C2×D3—362no—
422C2×D3—362no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—22no—
426C3×C4—41no—
427C3×C4—61no—
428C3×C4—61no—
429C3×C4—61no—
430C3×C4—61no—
431C3×C4—182no—
432C3×C4—182no—
433C3×C4#369712yes—
434C3×C4#380412yes—
435Dic3—21no—
436Dic3—21no—
437Dic3—21no—
438Dic3—21no—
439Dic3—62no—
440Dic3—62no—
441Dic3—182no—
442Dic3—182no—
443Dic3—362no—
444C32#553411yescharacteristic
445C9—21no—
446C9#564311yescharacteristic, complemented
447C23—272no—
448C23C2×D3×D911yescharacteristic
449Q8—182no—
450Q8—1082no—
451D4—542no—
452D4—1082no—
453C2×C4—22no—
454C2×C4—31no—
455C2×C4—31no—
456C2×C4—31no—
457C2×C4—31no—
458C2×C4—92no—
459C2×C4—92no—
460C2×C4—272no—
461C2×C4—1082no—
462C2×C4C2×D3×D911yescharacteristic
463C2×C4C2×D3×D911yescharacteristic
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—42no—
468C2×C3—41no—
469C2×C3—41no—
470C2×C3C2×#369711yescharacteristic
471C2×C3#6371511yescharacteristic
472C2×C3#6371611yescharacteristic
473C2×C3#6429711yescharacteristic
474C2×C3#6430811yescharacteristic
475C2×C3C2×#380411yescharacteristic
476D3—362no—
477D3—362no—
478D3—722no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—542no—
483C22—542no—
484C22—542no—
485C22C22×D3×D911yescharacteristic
486C4—21no—
487C4—61no—
488C4—61no—
489C4—61no—
490C4—61no—
491C4—182no—
492C4—182no—
493C4—542no—
494C4#8865812yes—
495C3—21no—
496C3#1058963611yescharacteristic, complemented
497C3#1059481411yescharacteristic
498C2—41no—
499C2—1082no—
500C2#1172246411yescharacteristic
501C2#1172247811yescharacteristic
502C2C2×#8865811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radical#11721819C2
hypercenterC22×C4D3×D9
hypocenterC3×C9#542
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C9×#926C2semidirect
3. C3×#63486C2semidirect
4. #11721275C2semidirect
5. #11721242C2semidirect
6. #11720994C2semidirect
7. #11720993C2semidirect
8. #11720970C2semidirect
9. #11720952C2semidirect
10. #11720920C2semidirect
11. Dic3×Dic18C2semidirect
51. #63486D3semidirect
153. #926D9semidirect
321. C3×C9#542semidirect
446. C9#5643semidirect
496. C3#10589636semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#117218192
C9×#9262
C3×#634862
#117212752
#117212422
#117209942
#117209932
#117209702
#117209522
#117209202
Dic3×Dic182
#105948143
#105896363